Lemma 36.27.4. Let S be a Noetherian scheme. Let f : X \to S be a flat proper morphism of schemes. Let E \in D(\mathcal{O}_ X) be perfect. Then Rf_*E is a perfect object of D(\mathcal{O}_ S).
Proof. We claim that Lemma 36.27.1 applies. Conditions (1) and (2) are immediate. Condition (3) is local on X. Thus we may assume X and S affine and E represented by a strictly perfect complex of \mathcal{O}_ X-modules. Since \mathcal{O}_ X is flat as a sheaf of f^{-1}\mathcal{O}_ S-modules we find that condition (3) is satisfied. \square
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